# Thabit number

An integer of the form $3\cdot 2^{n}-1$, or $2^{n+1}+2^{n}-1$. They are listed in A055010 of Sloane’s OEIS. The Thabit numbers are a subset of the Proth numbers.

The mathematician and astronomer Thabit ibn Qurra studied these numbers in search of a formula for amicable pairs. He found that when two consecutive Thabit numbers are also prime numbers (corresponding to indices $n$ and $n-1$) and $9\cdot 2^{2n-1}-1$ is a prime number, too, then these numbers multiplied by $2^{n}$ will reveal an amicable pair. The only $n$ known to fit these criteria are 2, 4 and 7. The largest Thabit number known to be prime corresponds to index 2312734, its immediate lower neighbor is composite.

It is conjectured that the nimfactorial of a Thabit number is always 2.

 Title Thabit number Canonical name ThabitNumber Date of creation 2013-03-22 15:52:58 Last modified on 2013-03-22 15:52:58 Owner Mravinci (12996) Last modified by Mravinci (12996) Numerical id 5 Author Mravinci (12996) Entry type Definition Classification msc 11A05 Synonym Thabit ibn Kurra number Synonym Thabit ibn Kurrah number Synonym Thabit ibn Qurra number Synonym Thabit ibn Qurrah number Synonym Thabit bin Kurra number Synonym Thabit bin Kurrah number Synonym Thabit bin Qurra number Synonym Thabit bin Qurrah number Related topic AFormulaForAmicablePairs